Simplify; express your answer in exponential form. Assume $z\neq 0, t\neq 0$. $\dfrac{{(z^{-4}t^{-2})^{3}}}{{(z^{-3}t^{-2})^{-2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-4}t^{-2})^{3} = (z^{-4})^{3}(t^{-2})^{3}}$ On the left, we have ${z^{-4}}$ to the exponent ${3}$ . Now ${-4 \times 3 = -12}$ , so ${(z^{-4})^{3} = z^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-4}t^{-2})^{3}}}{{(z^{-3}t^{-2})^{-2}}} = \dfrac{{z^{-12}t^{-6}}}{{z^{6}t^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-12}t^{-6}}}{{z^{6}t^{4}}} = \dfrac{{z^{-12}}}{{z^{6}}} \cdot \dfrac{{t^{-6}}}{{t^{4}}} = z^{{-12} - {6}} \cdot t^{{-6} - {4}} = z^{-18}t^{-10}$